Equation of a Plane in the Normal Form

The equation of a plane in the normal form can be determined if normal to the plane as well as the distance of the plane from the origin is known.

Equation of a Plane in the Normal Form

The vector form of the equation of a plane in normal form is given by:

Where $\vec{r}$ is the position vector of a point in the plane, n is the unit normal vector along the normal joining the origin to the plane and d is the perpendicular distance of the plane from the origin.

Let us say that P (x, y, z) is any point on the plane and O is the origin. Then, we have,

Now we take the direction cosines of $\hat{n}$ as l, m and n. So we have,

From the equation $\vec{r}.\hat{n}$ = d we get

Thus, the Cartesian form of the equation of a plane in normal form is given by:

lx + my + nz = d

Example

Let us take up an example to understand the equation of a plane in the normal form.

Problem: A plane is at a distance of $\frac{9}{\sqrt{38}}$ from the origin O. From the origin, its normal vector is given by 5$\hat{i}$ + 3$\hat{j}$ – 2$\hat{k}$.

What is the vector equation for the plane?

Solution: Let the normal vector be:

We now find the unit vector for the normal vector. It can be given by:

So, the required equation of the plane can be given by substituting it in the vector equation: